please solve it by easy way , and send clear picture . 2. Let Cla,히 be the space of continuous functions and define l|-lla via Show that (Cla, b),Il a) is a normed linear space. Moreover, prove...
please solve this question step by step and make it clear to understand. Also, please send clear picture to see everything clearly. thanks! 4. Let X be a set and C(X) be the space of continuous real-valued functions on X. Define Il llae by llfllx = sup If(z) 1. Prove that (C(X), Il . llo) is a Banach space. (You may assume that l-llx defines a norm on C(x)) 4. Let X be a set and C(X) be the space...
4. (10 points) Let X be the normed linear space of all simple functions in L(E). Show that X is not a Banach space. 4. (10 points) Let X be the normed linear space of all simple functions in L(E). Show that X is not a Banach space.
3. Let la, b) on [a, b]. Define | lo, by R and Cla, b be the space of continuously differentiable real-valued functions lsup () sup |f'() rEla,b rela,b Prove that (Cl a) is a Banach space 3. Let la, b) on [a, b]. Define | lo, by R and Cla, b be the space of continuously differentiable real-valued functions lsup () sup |f'() rEla,b rela,b Prove that (Cl a) is a Banach space
Consider the space V of continuous functions on (0, 1] with the 2-norm 12 J f2 We saw in class that V is an incomplete normed linear space. (a) For a continuous function p on [0, 1], define a linear map Mp: V-V by Mpf-pf. Show that Mp is bounded and calculate its norm. (b) Is A = (Mplp E C(0,1)) a Banach algebra? Note that B(V) is necessarily incomplete, so it is not enough to prove that A is...
please solve this question step by step and make it clear to understand .please send clear picture to see everything clearly. Thanks! 8. Let (Xy 11-11), j = 1, 2 be normed linear spaces. Prove that f : X1 → X2 is continuous if and only if f-(E°) C (f (E))o for every subset EX2. Here Eo denotes the interior of E.
Please solve this question. Sorry please neglect the bottom picture which says "moreover ...". I am happy to upbote if you solve (1)-(5). Problem 1. We denote by the set of all sequences (UK)x=1,2,... = (U1, U2, ...) (ux E C) u= satisfying luxl <00. Moreover, we define k=1 (u, v) = xox(u, v E f). k=1 (1) Prove that is a vector space. (2) Prove that is a inner product space with respect to (5.). (3) Construct the norm...
Problem 3. (1) Let H be a Hilbert space and S, TE B(HH). Then, prove that ||ST|| ||||||||| (2) Let X, Y be Hilbert spaces and Te B(X,Y). Then, prove that ||1||| sup ||T3|1 TEX=1 Let X, Y be Banach spaces. Definition (review) We denote by B(X, Y) a set of all bounded linear operators T:X + Y with D(T) = X. B(X, Y) is a vector space. Definition (review) A linear operator T:X + Y is said to be...
Please be more easy to understand,thanks! 14. Let 1gn) be a sequence of non-negative real-valued continuous functions defined on a closed interval [a, b]. Suppose that for each a E [a, b g monotonically, i.e., gn(x)0 and gn(x) 2 gn+1x)2... for all n E N (1) Prove that for each n E N there exists zn E a, b] such that n m)Mngn(): E [a,b) (3 Marks) (2) By contradiction, show limn-**o M ( n 0. (10 Marks) (3) Does...